Applications of representation theory and of explicit units to Leopoldt's conjecture

2023/01/13 by Ferri, Fabio, Johnston, Henri
#11R23 (Primary) 11R27 #19A22 #20C15 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2301.05700

Abstract

Let L/K be a Galois extension of number fields and let G=Gal(L/K). We show that under certain hypotheses on G, for a fixed prime number p, Leopoldt's conjecture at p for certain proper intermediate fields of L/K implies Leopoldt's conjecture at p for L. We also obtain relations between the Leopoldt defects of intermediate extensions of L/K. By applying a result of Buchmann and Sands together with an explicit description of units and a special case of the above results, we show that given any finite set of prime numbers P, there exists an infinite family F of totally real S₃-extensions of ℚ such that Leopoldt's conjecture for F at p holds for every F ∈ F and p ∈ P.

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